Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#2251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,12,16)(9,17,19,31)(10,22,24,37)(11,25,23,15)(13,26,27,20)(14,29,30,18)(21,36,35,33)(28,34,32,38);(7,9,18,34,11,21,24,26,12,19,29,38,23,35,10,20)(8,13,22,33,15,28,30,31,16,27,37,36,25,32,14,17);(3,4,5,6)(9,19)(13,27)(17,31)(20,26)(21,35)(28,32)(33,36)(34,38)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137542, #141556, #143213, … |
| maximal normal subgroup of | #137542, #141556, #143213, … |
| maximal quotient of | C2×#12507, C2×#12508, C2×#12512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26703);